Inequalities for pseudo arithmetic and geometric means

Abstract

We prove inequalities for the pseudo arithmetic and geometric means a _n and g _n defined by

$${a_n} = \frac{{{P_n}}}{{{p_1}}}{x_1} - \frac{1}{{{p_1}}}\sum\limits_{i = 2}^n {{p_i}{x_i}\quad and\quad {g_n}} = x_1^{{P_n}/{p_1}}/\prod\limits_{i = 2}^n {x_i^{{p_n}/{p_1}}} ,$$

where x _i and p _i (i = 1,...,n) are positive real numbers and ${P_n} = \sum\limits_{i = 1}^n {{p_i}}$. Further we prove a Ky Fan-type inequality involving the ratios a _n /a ^l _n and g _n /g ^l _n as well as an additive analogue involving a _n -a ^l _n and g _n -g ^l _n , where a ^l _n and g ^l _n will be obtained from a _n and g _n by replacing x _i by 1 - x _i with x _i ∈(0,1/2] (i = 1,..., n).